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In 5-dimensional geometry, there are 19 uniform polytopes with A5 symmetry. There is one self-dual regular form, the 5-simplex with 6 vertices.

Each can be visualized as symmetric orthographic projections in Coxeter planes of the A5 Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 19 polytopes can be made in the A5, A4, A3, A2 Coxeter planes. Ak graphs have [k+1] symmetry. For even k and symmetrically nodea_1ed-diagrams, symmetry doubles to [2(k+1)].

These 19 polytopes are each shown in these 4 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

# Coxeter plane graphs Coxeter-Dynkin diagram
Schläfli symbol
Name
[6] [5] [4] [3]
A5 A4 A3 A2
1 5-simplex t0.svg 5-simplex t0 A4.svg 5-simplex t0 A3.svg 5-simplex t0 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{3,3,3,3}
5-simplex (hix)
2 5-simplex t1.svg 5-simplex t1 A4.svg 5-simplex t1 A3.svg 5-simplex t1 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t1{3,3,3,3} or r{3,3,3,3}
Rectified 5-simplex (rix)
3 5-simplex t2.svg 5-simplex t2 A4.svg 5-simplex t2 A3.svg 5-simplex t2 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t2{3,3,3,3} or 2r{3,3,3,3}
Birectified 5-simplex (dot)
4 5-simplex t01.svg 5-simplex t01 A4.svg 5-simplex t01 A3.svg 5-simplex t01 A2.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t0,1{3,3,3,3} or t{3,3,3,3}
Truncated 5-simplex (tix)
5 5-simplex t12.svg 5-simplex t12 A4.svg 5-simplex t12 A3.svg 5-simplex t12 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t1,2{3,3,3,3} or 2t{3,3,3,3}
Bitruncated 5-simplex (bittix)
6 5-simplex t02.svg 5-simplex t02 A4.svg 5-simplex t02 A3.svg 5-simplex t02 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t0,2{3,3,3,3} or rr{3,3,3,3}
Cantellated 5-simplex (sarx)
7 5-simplex t13.svg 5-simplex t13 A4.svg 5-simplex t13 A3.svg 5-simplex t13 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,3{3,3,3,3} or 2rr{3,3,3,3}
Bicantellated 5-simplex (sibrid)
8 5-simplex t03.svg 5-simplex t03 A4.svg 5-simplex t03 A3.svg 5-simplex t03 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,3{3,3,3,3}
Runcinated 5-simplex (spix)
9 5-simplex t04.svg 5-simplex t04 A4.svg 5-simplex t04 A3.svg 5-simplex t04 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,4{3,3,3,3} or 2r2r{3,3,3,3}
Stericated 5-simplex (scad)
10 5-simplex t012.svg 5-simplex t012 A4.svg 5-simplex t012 A3.svg 5-simplex t012 A2.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t0,1,2{3,3,3,3} or tr{3,3,3,3}
Cantitruncated 5-simplex (garx)
11 5-simplex t123.svg 5-simplex t123 A4.svg 5-simplex t123 A3.svg 5-simplex t123 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,2,3{3,3,3,3} or 2tr{3,3,3,3}
Bicantitruncated 5-simplex (gibrid)
12 5-simplex t013.svg 5-simplex t013 A4.svg 5-simplex t013 A3.svg 5-simplex t013 A2.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,3{3,3,3,3}
Runcitruncated 5-simplex (pattix)
13 5-simplex t023.svg 5-simplex t023 A4.svg 5-simplex t023 A3.svg 5-simplex t023 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,2,3{3,3,3,3}
Runcicantellated 5-simplex (pirx)
14 5-simplex t014.svg 5-simplex t014 A4.svg 5-simplex t014 A3.svg 5-simplex t014 A2.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,4{3,3,3,3}
Steritruncated 5-simplex (cappix)
15 5-simplex t024.svg 5-simplex t024 A4.svg 5-simplex t024 A3.svg 5-simplex t024 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2,4{3,3,3,3}
Stericantellated 5-simplex (card)
16 5-simplex t0123.svg 5-simplex t0123 A4.svg 5-simplex t0123 A3.svg 5-simplex t0123 A2.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,2,3{3,3,3,3}
Runcicantitruncated 5-simplex (gippix)
17 5-simplex t0124.svg 5-simplex t0124 A4.svg 5-simplex t0124 A3.svg 5-simplex t0124 A2.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,2,4{3,3,3,3}
Stericantitruncated 5-simplex (cograx)
18 5-simplex t0134.svg 5-simplex t0134 A4.svg 5-simplex t0134 A3.svg 5-simplex t0134 A2.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,3,4{3,3,3,3}
Steriruncitruncated 5-simplex (captid)
19 5-simplex t01234.svg 5-simplex t01234 A4.svg 5-simplex t01234 A3.svg 5-simplex t01234 A2.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3,4{3,3,3,3}
Omnitruncated 5-simplex (gocad)


A5 polytopes
5-simplex t0.svg
t0
5-simplex t1.svg
t1
5-simplex t2.svg
t2
5-simplex t01.svg
t0,1
5-simplex t02.svg
t0,2
5-simplex t12.svg
t1,2
5-simplex t03.svg
t0,3
5-simplex t13.svg
t1,3
5-simplex t04.svg
t0,4
5-simplex t012.svg
t0,1,2
5-simplex t013.svg
t0,1,3
5-simplex t023.svg
t0,2,3
5-simplex t123.svg
t1,2,3
5-simplex t014.svg
t0,1,4
5-simplex t024.svg
t0,2,4
5-simplex t0123.svg
t0,1,2,3
5-simplex t0124.svg
t0,1,2,4
5-simplex t0134.svg
t0,1,3,4
5-simplex t01234.svg
t0,1,2,3,4

References

H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

External links

Klitzing, Richard. "5D uniform polytopes (polytera)".

Notes

Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

Mathematics Encyclopedia

Hellenica World - Scientific Library

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